**Algebra Preliminary Exam, Fall 1980**

- Suppose that for each prime integer
*p*dividing the order of a finite group*G*, there is a subgroup of index*p*. Prove that*G*cannot be a nonabelian simple group. - A subgroup of a finite group is ``
*p*-local" if it is the normalizer of some Sylow*p*-subgroup. Show that the number of*p*-local subgroups of a group is congruent to 1 modulo*p*. - Characterize (with proof) all finitely generated -modules with the property that each submodule is a direct summand. ( denotes the field of rational numbers.)
- Let
*R*and*S*be local Noetherian integral domains with maximal ideals*M*and*N*respectively. Assume that and that*S*is a finitely generated*R*-module. If there exists a proper ideal*I*of*R*such that and the canonical image of*R*/*I*in*S*/*IS*equals*S*/*IS*, then prove that*R*=*S*. - Let
*R*be an integral domain. For , define . Let be a set of prime ideals of*R*with the property that if and , then for some . Prove that . ( denotes the localization at .) - If
*G*is a finite group, prove that there exist fields*K*and*L*such that*L*is a Galois extension of*K*with Galois group isomorphic to*G*. - Show that for each positive integer
*n*, there is a polynomial such that for each matrix*A*with complex entries,( denotes the field of complex numbers and denotes the trace of

*A*.)

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Thu Aug 1 09:44:57 EDT 1996